lege, relege, labora et invenies

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Systèmes linéaires published 2006 in France by Hermes Science/Lavoisier © LAVOISIER 2006

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Library of Congress Cataloging-in-Publication Data

Bourl–s, Henri.
  [Syst–mes lin–aires. English]
  Linear systems / Henri Bourlès.
    p. cm.
  Includes bibliographical references and index.
  ISBN 978-1-84821-162-9
 1. Linear systems. I. Title.
  QA402.B62713 2010
  003'.74--dc22

2010016973

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-162-9

Preface

The notion of system

The notion of system is the basic concept of control theory. What is a system? It is quite difficult to address this question in all its aspects. We can say somewhat loosely that it is an entity consisting of interacting parts; an entity that is itself, most often, also interacting with other systems. The solar system, a computer system, etc. are examples of a system.

The control system analyst is interested in systems which are, at least in part, designed and constructed by man, in order to be utilized – a car engine or the entire automobile; an alternator in a power station or the “power system” in its entirety (consisting of production centers, lines of energy transport and consumption centers); an airplane, such as the A380 Airbus, where control systems play a very important role; a factory production line, etc. We can act upon these systems and these react to actions exerted on them.

Objectives of systems theory

Several objectives exist in control systems theory.

Modeling

The above-mentioned systems are part of the material world and are thus governed by the laws of physics. By putting the system in an equation based on these laws, we obtain a mathematical model which will greatly facilitate its understanding. Therefore, modeling is one of the important activities of the control systems analyst. The modeling process does not always start with the laws of physics; it can simply be based on a more or less empirical and qualitative observation of the behavior of the system. Within the framework of this book, we will however limit ourselves to cases where a mathematical description of the system behavior is possible. In general, the model obtained consists of a set of differential equations (sometimes of partial differential equations) or of difference equations.

Identification

The modeling process, as understood, involves determining the structure of the equations which govern the behavior of the system, and also in fixing a priori the values of certain system parameters: for example, the lengths, masses, resistance values, capacitance values, etc. But it is often impossible to come to an a priori complete and precise understanding of all the parameters of this model. In order to refine and complete this understanding, it becomes necessary to proceed to an identification of the system: from the reactions of the latter to known and given stimulations, we can, under certain conditions, identify the yet unknown parameters. Identification is one of the major aspects in control theory.

Analysis

Once the system is modeled and identified, it becomes possible to analyze its behavior. This analysis can be very complex. As an example, the analysis of the European and North American electric systems is difficult and requires powerful computing resources and enormous databases because we deal with very large systems. And understanding these systems well is essential for security reasons: points such as the possibility of “black-out” risk? In general, the analysis of a system makes it possible to determine its essential properties.

Control

Thus, systems theory is a “theoretical” science in which one of its objectives is knowing systems, through modeling, identification and analysis. But it is also (we may be tempted to say essentially) a “practical” science, a science of “action”. We try to understand systems in order to be able to control them, and to regulate them in the best possible way. The last major chapter of systems theory is that of control.

Open-loop and closed-loop systems

A fundamental difference exists between “open-loop” and “closed-loop” systems. Controlling an open-loop system is doing it blindly, without taking into account the results of the action taken. We know the expression: “I could walk down this path blind-folded”. Because I know at which moment I need to turn left, and then right. I know when I need to accelerate, when to slow down, . . . And it is true that if, as with Laplace–s genius, we had a perfect knowledge of the world, we could control, in open-loop, every system belonging to it. However, our knowledge of things is incomplete. Even with a route we know by heart, unpredictable events may occur: a child unexpectedly crossing the street, a spell of rain making the road more slippery than usual, etc. It is thus necessary to note at all times, and to adjust actions, taking into account the reality that appears from moment to moment. Control theory is the art (or science) of making this unceasing adjustment, which we call a loop or a feedback or a servo-mechanism. It is a common concept which nevertheless poses numerous problems.

One of the major difficulties encountered with feedback systems is their possible instability. There are certainly unstable open-loop systems, but they are relatively rare: a helicopter, an alternator connected with a long power line, certain types of fighter jets, etc. On the other hand, nothing is more commonplace than making a system unstable with a poor-performing feedback loop. Let us think of a child striving to take a shower: when the water is cold, he wants to heat it up, turns the hot water tap on too quickly, and gets boiling hot water; then, over-doing it the same way with the cold water tap, he makes the water ice cold, and so on.

The notion of feedback control is thus very powerful, but cannot be applied without proper knowledge.

Presentation

This book is divided into chapters, and sections; for example, section 2.1 is the first section of Chapter 2, and section 2.1.3 is the third subsection of section 2.1

It contains the basics that a non-specialized engineer must know in control engineering. I had the opportunity to teach this course at Conservatoire National des Arts et M–tiers (Paris), several engineering institutes Grandes Ecoles, Ecole Normale Sup–rieure de Cachan, and, for the most difficult parts, at Paris XI University (Master 2 level). The course contained in this book is progressive, making it accessible to any reader with an L2 level in science, and includes many examples. Nevertheless, to make it coherent, passages (sections or groups of sentences) had to be included in the first chapters that call upon somewhat more difficult notions which may need a second or third reading. These passages are preceded by asterisks for sections, and situated between two asterisks for groups of words or sentences.

The purpose of this book is to study system modeling, identification, analysis and control. Among these four themes, modeling is perhaps the trickiest problem: to know how to model electrical, mechanical, thermal, hydraulic, or other systems – as they are complex – a person has to be an electrical, mechanical, thermal, hydraulic engineer. It is physics in general, all of physics, which is useful for modeling; and modeling, which is the subject matter of Chapter 1, is not exclusive to the control engineer. Some examples will serve as basic reminders of hydraulics and thermodynamics. Reminders relative to electricity are somewhat less succinct. With respect to mechanics, I thought that it would be useful to go into more details. However, the only objective of the presentation is to enable the reader to understand examples and resolve exercises. Obviously, it cannot replace a treatise on mechanics.

From Chapters 2 to 11, the control engineer finds himself/herself in his/her own private domain. These chapters deal with linear systems analysis, control, and identification. I have put together some mathematical elements which I deemed essential in the appendices included in Chapters 12 and 13. This will spare the reader from constantly referring to the bibliography: elements of analysis first, and then of algebra. Some of them (Smith form of a polynomial matrix, images algebra, module theory etc.) are probably new to most readers. But I preferred not to discuss them in the main body of the text, to save the latter, to the extent possible, for what truly comes within the scope of control theory. Likewise, the Laplace transform and the z-transform, which are mathematical tools that classically appear high in textbooks on control theory, are included in these appendices. These are organized as a presentation of “mathematics for systems theory” with proofs when they are constructive or simply useful for the reader–s understanding. The reader can thus choose to refer to the appendices if needed or to read them entirely as actual chapters. I have decided not to present the theory of measure and integration. Concerning the latter, I refer the reader to the second volume of [35] or to [103]. I achieved this with the help of some mathematical gymnastics here and there.

About 80 exercises are given to the readers to test and refine their understanding of the subject. Solutions to most of the exercises are provided (sometimes in brief) in Chapter 14.

MATLAB and/or SCILAB files to run examples and solve exercises can be found at www.iste.co.uk/bourles/LS.zip.

Course outline

Beginner–s course

The beginner at L3 (i.e. third-year undergraduate) level can start with the easiest parts of Chapter 1. The part that might be difficult for the beginner is section 1.2 which is devoted to the modeling of a mechanical system. I advise the reader to approach this part in a pragmatic way, using the examples and exercises which show what results are essential and how to make good use of them.

In Chapter 2, readers should leave aside everything that relates to multi-input multi-output systems. It is better if they focus on “left-forms”, “right-forms”, and on methods which make it possible to obtain system transfer functions. They will eventually develop particular interest in treated examples and exercises.

The reader should study Chapters 3 to 6 in depth (leaving aside, of course, the starred passages “intended for a second reading”). From my point of view, the reader–s course therefore ends with the RST controller (“usual case” section).

Advanced course

The reader at M1 (i.e. first-year graduate) level should begin to go deeper into what he/she only skimmed over when he/she was a beginner, i.e. section 1.2 of Chapter 1, and Chapter 2. Afterwards, the reader will resolutely move on to Chapter 7, which includes complementary material on systems theory, and will then be ready to study Chapters 8 to 11. I advise the reader to systematically skip the passages related to module theory.

Advanced graduate level

The reader at M2 (i.e. second-year graduate) level still has many things to discover before launching into specialized courses. These are all the passages that he/she skipped during previous readings, in particular those presenting the theory of systems in the language of modules. It may also be in the interest of the readers to systematically re-read the elements of mathematics in Appendices 1 and 2 (Chapters 12 and 13).

Coherent course and “butterfly” course

This book should be read in a more systematic way to understand the contents. Another approach to this book is also possible for readers who wish to be systematic while not necessarily belonging to the above categories: these readers should read the chapters of the book, without skipping any passage, in the order I have presented them (with the exception of the appendices – Chapters 12 and 13 – which they should read first). I have often started reading a science book in that spirit, and only a lack of time has made me take an opposite turn, which is to “flit about”, trying to go as fast as possible (a strategy, besides, which does not always pay off). I do not have any way to propose to the “butterfly” readers, but I have included a detailed index at the end of the book for them.

Choices and necessities

Since this book is about linear time-invariant systems, I have deemed it essential to present this concept correctly. Such a system can no longer be properly defined by its transfer matrix (too poor a representation), or by Kalman–s approach, as one of its state realizations, or by Rosenbrock–s approach, as a “system matrix” (too contingent a representation). Wonham–s “geometric approach”, which was proposed in the 1980s, in the end, only reformulates Kalman–s approach in a little more modern mathematical language, and does not seem to be a good answer. Following the works of experts in differential algebra, it appeared that, what control engineers call “system” and “linear system”, should be defined as an extension of a differential field and a module defined over a ring of differential operators, respectively. In the early 1990s, this property was revealed by M. Fliess to the community of control engineering (see the synthesis presented in [47]). Such a language may have discouraged a beginner, but it is this conception that I have attempted to make the reader gradually sensitive to, in Chapters 2 and 7. In reality, module theory, at the level used here (finitely generated modules over principal ideal domains) is very simple for a reader somewhat accustomed to abstraction. It does clarify manipulations that can be done with polynomial matrices, as well as the Jordan canonical form. For this reason, I have made it the main theme of “passages intended for a third reading”. The reader wishing to go further into systems theory can refer to [47] on nonlinear aspects, and [22] on linear time-varying systems ([15] is a preliminary version of [22]). Module theory is helpful in presenting (in the case of linear systems) the notion of flatness (due to Fliess and his collaborators [48]), which has become essential to efficiently resolve the problem of motion planning (see section 7.5).

I wanted to show that control engineering is not magic but science. On that premise, I have insisted, in Chapter 4, on the limitations due to certain characteristics of the system to be controlled (unstable poles and zeros, specifically). For more details on this point, see [51].

The control engineer is permanently confronted with the problem of uncertainties in modeling. Thus, this book is constantly preoccupied with the robustness of control laws. However, this is not a book on robust control. The theory of robustness has become extremely sophisticated, and its recent development is not covered here. Besides, excellent books on this issue ([107] and [122], among others) do exist.

I have emphasized in this book on control (whether in polynomial or state formalism) that system stabilization is exceptionally the role of the feedback loop; the point is often to compel the output of such a system to follow a reference signal, in spite of various disturbances that it may be subjected to. This is why – in order to design an RST controller as well as a control by state feedback, possibly with an observer – it is essential to first make the necessary model augmentations (following what Wonham called the “internal model principle”). Without this prerequisite, “modern methods” only generate nonsense.

It also appeared to me that it was important to show that, in the multi-input multi- output case, a control law with execrable qualities can correspond to a good pole placement, because the latter does not determine the former. The worst method (even though it is quite elegant from a strictly theoretical point of view) is probably the one that consists of reverting to a cyclic state matrix through a first loop, and then proceeding as in the single-input single-output case.

I have abstained from presenting the linear-quadratic (LQ) optimal control and its “dual” version, the Kalman filter. For the conception of control by state feedback and observer, I have indeed preferred to propose essentially algebraic methods for several reasons. Once these methods are well assimilated, it becomes easy to implement an efficient “LQG control”.¹ The theory of LQG control (which encompasses LQ control and Kalman filter theories) is now classic and, at a basic level, is well covered in treatises such as [1] and [2]. On the other hand, the minimization of a criterion is only a convenient intermediate step for the control engineer to obtain a control law with the desired properties [78]. Another element that kept me from including optimal control is that it would have made this book longer (especially to correctly present continuous- time stochastic optimal control, with, in the background, Ito–s differential calculus [68], [118]). I found the volume of this book to be large enough. Nevertheless, except for a few subtleties, the methods proposed in the multi-input multi-output case in Chapters 8 and 9 only differ from LQ control and Kalman filtering in their presentation style (the minimization of a criterion is not presented as a goal – see Remark 245 in section 8.1.4). I encourage the reader to continue the study of this book along with that of “LQG control with frequency-shaped weights” [2], which is a good complement.

Only Chapter 10 deals with discrete-time systems and control. All previous chapters (including the one on RST controller) are thus presented in the context of continuous-time, which is a bit unusual. Despite the ideas spread by some people, I have experienced that continuous-time formalism offers much more flexibility for the design of control laws, especially with respect to the choice of poles and the question of “roll-off”, which are essential to robustness. Once the synthesis of continuous-time control laws is mastered, it is very simple to switch to the synthesis of discrete-time control laws – without any approximation (this point is of course discussed in detail in Chapter 10).

Chapter 11 discusses parametric identification of discrete-time systems by minimization of the l₂ norm of the prediction error. The presentation is relatively classic as far as open-loop identification is concerned; as for closed-loop identification, it essentially includes Ljung and Forsell–s contribution [82] (as a complement, the reader can consult [75] and Chapter 10 of [110]).

Many other topics would have been worth presenting: for example, anti-windup methods in the presence of saturations (which is clearly expounded in [4]) or gain scheduling (see survey papers [104] and [80]), not to mention nonlinear control (for a general presentation of this theme, see [108] and [62], and begin with the first reference because it is simple; robust nonlinear control is discussed in [49]; see also [48] which deals with flatness from both theoretical and practical viewpoints, in addition to section 7.5 of this book). We can also cite the following subjects: adaptive control (which is the subject of a lot of literature, but [57], [3] and [108] can be recommended as a good initiation on this theme); fuzzy or neural control (the reader can find a genuinely scientific presentation of the last two types of control in [40]); the control of systems governed by delayed differential equations or partial differential equations (on this subject, see [55] and [33] in the linear context, as well as [58] and [59] where nonlinear problems are discussed). As already mentioned, the extension of many methods presented in this book to linear time-varying systems can be found in [22].

Note on the English edition

This English edition has given me the opportunity to correct many errors which, despite proof-reading, had been left in the original French edition, and also include additional information which might be useful. Furthermore, it has given me the opportunity to complete Chapter 7 with a section on flatness and to entirely revise Chapter 11 on identification: the mathematical bases included therein are given in Appendix 1, and a section on closed-loop identification has also been added. Finally, this edition contains additional exercises and one of them presents the basics of what needs to be known about the “delta transform”.

Acknowledgments

I would first like to thank my “elders”, especially E. Irving, I.D. Landau and P. de Larminat, who have shared their experience with me during informal, but always passionate, discussions. I also thank M. Fliess who has opened my eyes to the algebraic theory of systems – which I hope this book has benefited from – and P. Chantre for our exchanges on identification. I thank my son Nicolas who has helped me finalize figures. I also thank my translator G.K. Kwan for his efforts and his kindness. Last but not the least, I thank my wife Corinne, whose patience has often been severely tested during this work which has demanded a lot of time and energy from me.

1. “LQG” stands for “Linear Quadratic Gaussian”.

Chapter 1

Physical Models

1.1. Electric system

Electric circuits can be modeled by applying Kirchhoff’s two laws: Kirchhoff’s Current Law and Kirchhoff’s Voltage Law. They are also known as the nodal rule and the mesh rule. Following are the two examples in which these two laws have been applied.

1.1.1. Mesh rule

Consider the “RLC” circuit shown in Figure 1.1.

This circuit constitutes a mesh. The mesh rule states that between any two points on a mesh, points A and B in this circuit for example, the potential difference is independent of the path that is taken (which, in physics, is a fundamental property of a potential, in this case the electric potential). Therefore, we have Va – Vb =V  = Vr + VL +VC, where Vr,VL, and VC are the voltages across the resistance R, the inductance L, and the capacitance C, respectively. We have VR = Ri, VL = images therefore we obtain the integro-differential equation. images Differentiating this equation we get the following linear differential equation:

(1.1) images

Figure 1.1. RLC circuit

1.1.2. Nodal rule

Consider the electric circuit given in Figure 1.2, which is a “pi circuit” used to represent a transmission line in the domain of electric networks.

According to the mesh rule described above, we have

The nodal rule is based on the conservation of charge. According to this rule, at any node of a circuit, which can be any point within the circuit, the sum of current entering a node is equal to the sum of current leaving that node. Hence, we have the following two equations: iA= i + i₁ and i = iB + i₂. Eliminating the variable i, we can now re-arrange the equations as follows:

(1.2) images

Figure 1.2. Pi circuit

1.2. Mechanical system

There are two main approaches to model a mechanical system. The first one uses the fundamental principle of dynamics, and is based on the analysis of forces and moments applied to the various elements of the system (particularly, internal forces, i.e. forces applied by subsystems on one another). The second approach uses the Lagrangian formalism and is based on the energy of the system (kinetic and potential energy). The analysis of internal forces is then unnecessary. These two approaches are succinctly expounded and applied to two examples.

1.2.1. Fundamental principle of dynamics

Torsor

A torsor is a field of antisymmetric vectors in a three-dimensional affine space. Let images be a field of such vectors; this is a function images is called the moment of the torsor images at point A. There exists a vector V, called the characteristic vector, such that for any points A and B, we have the relation images The torsor is thus entirely determined by images and the moment images for any point A under consideration. We are agreed on the notation images for the pair images and images for the torsor images . The two vectors imagesandimages are called the elements of reduction of torsor images at point A.

The comoment of images is the scalar images This comoment is invariant in the sense that it is independent of the point A considered (as can easily be verified by the reader). Therefore, the comoment of the torsors images and images is well defined and is denoted by imagesimages

Kinematic torsor

An example of a torsor is the field of velocities of a rigid body S, called the kinetic torsor of S; this torsor is defined relative to an orthonormal frame of reference images (we also say the torsor is defined in this frame of reference). This is to say that the translational velocities and angular velocities are measured relative to the origin and along the axes of images .Its characteristic vector is the vector of instantaneous rotation” images (this torsor is thus denoted by images Indeed, as easily shown, the velocities at two points A and B on this rigid body are related by

(1.3) images

Now, let us consider a frame of reference images attached to the rigid body S. This consists of an origin A and three basis vectors images where Ai’s are points on S. We have images according to (1.3), and thus images be any vector; its coordinates within images are those Yi ’s such that images Differentiating this expression, we get

This quantity, denoting it by images is the derivative of images in the reference frame images The vector images is the derivative of images in this moving reference frame images and is denoted by images. We have thus obtained the formula establishing the relation between the differentiation in a moving reference frame and the differentiation in a fixed reference frame as follows:

(1.4) images

Kinetic torsor

Kinetic moment

The kinetic moment of a material system S (i.e. a collection of material points and rigid bodies – a material point can be considered a punctual rigid body) with respect to any point A in reference frame images is

where dm is the density of mass at point M; this density is assumed to be constant over time. ¹

Elements of reduction

The field of vectors images is a torsor, as will be shown below, and it is called the kinetic torsor.

Indeed, if B is any other point, we have

Now let O be the origin of images we then get images = images By definition of the center of mass G of S, we have

where m is the total mass of S. Therefore,

where images is the momentum of S, defined as

According to (1.5), images is actually a torsor images with characteristic vector images

Case of a rigid body

Suppose S is a rigid body and A is any point on this rigid body; according to (1.3) we get images thus

The linear mapping images dm is called the inertia tensor of S with respect to A, and hence we have

(1.6) images

We can express this tensor in terms of its coordinates in an orthonormal system of reference images Let (x, y, z) be the components of images in imagesDeveloping expression (1.6), the linear mapping images is identified² with the inertia matrix of S with respect to A in images given by

We thus obtain

(1.7) images

This expression can be simplified when A = G or when A is a fixed point on S (if S rotates around this point); then (1.7) reduces to

(1.8) images

Inertia matrix

Matrix images is only constant when the reference frame images is rigidly linked to rigid body S and it is therefore our interest to look at things in the context of this case. Of course, this reference frame is in motion, and the derivative of (1.8) is obtained by applying (1.4). On the other hand, the matrix images is symmetric real, and thus can be diagonalized in an orthonormal reference frame (see section 13.5.6), and whose axes are by definition the principal axes of inertia of S. The diagonal elements obtained in such a matrix are called the principal moments of inertia of S (with respect to A and relative to the principal axes in question).

Torque

In the case where images, the kinetic moment is independent of the point being considered, according to (1.5): for any points A and B, images. Such a kinetic moment is called a torque, which we shall denote by images

Kinetic energy

The kinetic energy T of a material system S is equal to the comoment images. In case of a rigid body, we obtain

(1.9) images

Force torsor

Now consider a set of external forces images being applied to points A₁,…, An of a material system S. The resultant force is given by images while the resultant moment of these forces with respect to a point O, arbitrary at this time, is images If A is another arbitrary point, we immediately obtain the equality

(1.10) images

which shows that the field imagesdefines a torsor with characteristic vector images this torsor images is called the force torsor (or, more precisely, the torsor of external forces).

Fundamental principle of dynamics (Newton’s law)

Expression in a Galilean reference frame

Let O be a fixed point in a Galilean reference frame (also known as inertial reference frame) images and S be a material system. The fundamental principle of dynamics or Newton’s law is written as

(1.11) images

This equality between torsors shows that we have the following two relations

(1.12) images

(1.13) images

It is reasonable to emphasize the fact that (1.11) is valid in the reference frame 1Z (or in any other Galilean reference frame). In a non-Galilean frame of reference, it is necessary to take into account the torsor of inertial forces as well as Coriolis forces [76] – notions that can be derived from (1.4).

Expression in a moving frame of reference firmly fixed in the center of mass

Nevertheless, let us consider the center of mass G of the material system S; from (1.5) we get

according to (1.12). Therefore,

which shows that

Expression in an arbitrary moving frame of reference

Let A be any arbitrary point. We have, according to (1.8), images, and thus

From (1.10) and (1.12) we also have

(1.14) images

From (1.13) we obtain

which is a generalization of (1.13).

Moving frame of reference: case of a rigid body

Let A be a point on rigid body S. The kinetic moment images is given by expression (1.7), therefore

According to (1.14) we obtain the following:

(1.15) images

This expression can be simplified in the following cases:

i) A is fixed and S rotates around A, or A = G (the second term on the right-hand side drops out);

ii) rotation is around one of the principal axes of inertia (the third term drops out).

Case of a rigid body rotating around an axis

Consider the case where S is a rigid body rotating around one of the principal axes of inertia, denoted by images for example. Let images be the resultant torque exerted on S (directed along images) and write images = images, where images is the unit vector of images The rotation vector of S is images We have images , where J is the principal moment of inertia of images with respect to the axis imagesi.e.

(1.16) images

Equation (1.15) reduces to

(1.17) images

Principle of action and reaction

In the case of a material system S that consists of N interacting rigid bodies Si, the global equation (1.11) is not sufficient to determine the motion of each of the rigid bodies. We therefore are led to decompose the system into each of its elements Si and to take into account the forces and moments these rigid bodies exert on one another.

Figure 1.3. Train

According to the principle of action and reaction, if images and images represent the torsor of the forces exerted on Sj by Si, and on Si by Sj , respectively, we have

(1.18) images

Example of a frictionless train

Consider a train composed of a locomotive and a wagon such as depicted in Figure 1.3. The equations of this system are formulated with the help of (1.12) and (1.18).

The coupling between the locomotive of mass m₁ and the wagon of mass m₂ is represented by a spring with constant k (according to Hooke’s law). The positions of center of mass of the locomotive and that of the center of mass of the wagon are denoted by z₂ and l+z₁, where l is the distance between the two centers of mass when the train is at rest. The motional force put into action by the locomotive is denoted by f. The frictional forces are neglected along with the mass of the spring. The resultant force exerted on the locomotive is therefore f -fr , where fr is the force exerted by the spring on the locomotive. As a result, we have images with z₁ -z₂ being the lengthening of the spring. The spring transmits the integral force fr according to (1.12). The wagon is only subject to this force fr and we thus have images Eliminating the variable fr, we obtain the equations

(1.19) images

Example of the inverted pendulum

Consider the inverted pendulum in Figure 1.4. It consists of a carriage with which a rod of length l terminated by a mass m is articulated. The frictional forces as well

Figure 1.4. Inverted pendulum

as the mass of the rod are considered negligible; in addition, mass m is considered punctual.

The inverted pendulum is acted upon by a horizontal force f that is exerted onto the carriage.

First, let us examine the forces that are exerted on the carriage. Since the carriage moves horizontally, only the horizontal components of the forces have to be considered. The horizontal resultant force on the carriage is f – fr, where fr is the horizontal component of the force exerted by the rigid body (rod + mass m) onto the carriage. According to (1.12) we thus have images

Now, let us examine the rigid body (rod + mass), whose center of mass G has coordinates (y1, z1). The only horizontal force exerted on this rigid body is fr, henceimages

We have the geometrical relations

(1.20) images

By way of the first of these equations and by eliminating fr, we obtain

We can write equation (1.15) at the point A. The term images is zero, since the rotation is effected around one of the principal axes of inertia. Therefore, (1.15) is written as images and the second term on the right-hand side can be interpreted as the moment with respect to A of the inertial force images applied at G. The other moment to be accounted for is that of the gravitational force. We thus obtain images cos Ɵ, where J is the moment of inertia of m with respect to A, i.e. ml² .

Finally, we obtain

1.2.2. Lagrangian formalism

For certain types of mechanical systems, it may be more convenient to use the Lagrangian formalism instead of the fundamental principle of dynamics, for the formulation of the equations. As mentioned above, one of the advantages is that the internal forces of the system do not have to be considered.

Holonomic system

A mechanical system S is said to be holonomic if the position of its different parts can be characterized by n independent variables q₁,…,qn, called the generalized coordinates of the system. We then say that S is a holonomic system with n degrees of freedom.

For example, the train above is a holonomic system with generalized coordinates q1 = y1 and q₂ = y₂. The inverted pendulum is also a holonomic system with generalized coordinates qi = y and q₂ = θ.

The vector space of dimension n in which these coordinates are defined is called the configuration space.

Holonomic relations

The variables q₁,…,qn are only independent if all existing relations between the subsystems have been taken into account and all redundant coordinates have been eliminated.* For example, suppose that, when all relations are removed, the system S only depends on N coordinates q₁,…, qN. Let us also assume that the only relations by which S is constrained are p relations of the form

(1.22) images

where gk. are scalar functions. Relations of the form (1.22), involving only the positions qi, are called the holonomic relations. These relations are of a pure geometrical nature. Let n = N — p and suppose that the Jacobian images is non-zero. Then, according to the Implicit Mapping Theorem, we can express (locally) qn +1 …, qN as a function of q₁,…, qn which are independent variables. In this case, S is a holonomic system and qN,…, qn are its generalized coordinates.*

Non-holonomic relations

The situation becomes more complicated in the case where the relations involve the velocities q₁ …, qN, according to hj (.q₁,…, qNv, q₁ …, qN) = 0. These relations, of kinematic (not geometric) nature, are called non-holonomic. Then, the system S itself is also called non-holonomic. This situation will not be considered in the sequel.

Kinetic and potential energies; external forces and torques

Let S be a holonomic system with n degrees of freedom. Its kinetic energy is the sum of the kinetic energies of its subsystems; it depends on the time derivatives of the generalized coordinates and very often depends on these coordinates themselves. For each part of the rigid body S, it is given by (1.9). The kinetic energy of S is denoted by T (q1, …,qn,q₁ …,qn).

In the same manner, the potential energy of S is the sum of the potential energies of its subsystems. This energy only depends on the generalized coordinates; it is denoted by U (q₁,…, qn).

Moreover, S can also be subjected to external forces and torques, i.e. which are not derived from the potential U (whether we have chosen not to take them into account in U, or whether they cannot be derived from any potential). So, the force f has to be treated as an external force in the example of the train and that of the inverted pendulum.

Generalized forces

Consider the external forces and torques exerting on a holonomic system S. While one of the generalized coordinates qi varies with an infinitesimal quantity δqi, and while the other generalized coordinates remain constant, an infinitesimal work δWi = Qi δqi is done as a result of these external forces and torques. Then Qi is said to be the (external) generalized force in the ith direction of the configuration space – or, in short, the generalized external force along qi. In the example of the train, f is the generalized external force along y₁, whereas in the example of the inverted pendulum, / is the generalized external force along y.

Lagrangian equations

The Lagrangian of the system is

(1.23) images

and the Lagrangian equations are

(1.24) images

whereimages

Example of the train

We have images The potential energy is that of what is stored inside the spring, and is given by images We can derive (1.19) from (1.24).

Example of the inverted pendulum

The kinetic energy of the carriage is images and that of the assembly (rod + mass) is images. By way of the holonomic relations (1.20), we obtain

The potential energy is that of the weight, which is given by U = mgl cos θ. Applying (1.24), we immediately obtain (1.21).

1.3. Electromechanical system

A lot of systems consist of an electrical and a mechanical part. This is the case, for example, of an alternator connected to an electric network [70]. This is also the case of a synchronous motor [112] or that of an asynchronous motor [24]. The modeling of these systems requires too much circuit theory and techniques to be covered in this work. We will content ourselves with considering only the case of a DC motor [112], even though there seems to be a trend where the DC motor is being replaced by the motors we just mentioned.

The windings of the rotor are connected to the armature by rings and brush. The stator generates a fixed field (either using permanent magnets or using windings in which a DC current is flowing) wherein the turns of the rotor are rotated with angular speed ω. With a voltage V applied to the armature, a current i passes through these turns. The charge is characterized by its moment of inertia J (see Figure 1.5).

Figure 1.5. DC motor

According to the mesh rule,

where e is the back electromotive force. According to Faraday’s law, images where images is the magnetic flux in the windings of the rotor. This flux is constant with respect to the stator, and varies in proportion to ω and with respect to the rotor. We can then let e = K ω; the coefficient K ≠ 0 is called the motor constant. The power exchanged between the stator and the rotor is P = e i (in electrical terms) or P = Cm ω (in mechanical terms). Therefore, Cm ω = K ωi, from which we deduce that Cm = K i. Finally we get, according to (1.17),

where Cr is the resistive torque, due to friction. This resistive torque is of the form Cr = λω, where λ is the frictional coefficient. In addition, the angular position θ of the rotor satisfies the equation images We can now regroup the equations in the following form:

(1.25) images

1.4. Thermal hydraulic system

Thermal or hydraulic systems are largely formulated using relations of balance: conservation of mass, conservation of energy. “Matter can neither be created nor destroyed, only transformed”. This formula of Lavoisier is undoubtedly the most universal principle of science.

Consider, for example, the heated tank in Figure 1.6.

This tank consists of a reservoir with section S. It contains water which is heated by a resistance R. Qe is the rate of discharge of water (at a constant temperature Te) entering the reservoir. The tank is emptying through an opening of cross-section a located at the bottom of the reservoir; rate of water discharging from the tank is denoted by Qs. The temperature of the water inside the tank is assumed to be uniform and denoted by Ts. The heating power is P.

1.4.1. Balance in volume

For an infinitesimal interval of time dt, let dV be the increase in volume of water inside the reservoir. We have dV = S dh, where dh is the increase in height

Figure 1.6. Heated tank

of water. The volume of water entering the reservoir during this time interval Qe dt. And the volume exiting is Qsdt. We have therefore (under the assumption o incompressibility)

1.4.2. Exit rate: Torricelli’s formula

This formula gives velocity vs of water exiting the reservoir under height of water h:

where g is the gravitational acceleration. Since Qs = σ vs, we obtain images

1.4.3. Energy balance

Let dW be the amount of electric energy furnished during the infinitesimal time interval dt. Let also dWe be the calorific energy (i.e. heat energy) furnished by the liquid entering the tank, and dWi be the increase in calorific energy in the interior of the tank that produces variation of the temperature Ts. According to the first principle of thermodynamics, or principle of conservation of energy:

On the other hand, dW = P dt. Denoting by /x the mass of the volume of liquid and by c the specific heat, we have dWe = Qe μc (Te —Ts)dt and dWi = Sh μ c dTs.

Figure 1.7. Electric circuit

Re-arranging the equations, we obtain

(1.26) images

1.5. Exercises

EXERCISE 1.- Consider the electric circuit in Figure 1.7. Determine the differential equation relating current i to voltage V.

EXERCISE 2.- Consider the double inverted pendulum as shown in Figure 1.8. Determine the equations of its motion.

EXERCISE 3.- [72] Consider the mixer given in Figure 1.9.

This mixer consists of a reservoir that receives two liquids of constant concentrations c₁ and c₂ and whose discharge rates Q1 and Q₂ are regulated by means of valves. The liquid is stirred within the reservoir, for concentration cs to remain uniform. The mixture exits through an opening of cross-section a located at the bottom of the tank. Determine the equations of this system.

Figure 1.8. Double inverted pendulum

Figure 1.9. Mixer

EXERCISE 4.- We consider another mixer, but this time the hot water at temperature T₁ enters the tank with rate Q₁, and the cold water with temperature T₂ enters with rate Q1. (It is a bathtub.) The temperatures T₁ and T₂ are fixed. Water flows to the bottom of the tub where we have omitted to plug the opening having cross-section σ; temperature of water inside the tub is T₈. Determine the system equations. What are the connections between this exercise and Exercise 3?

1. The symbol images means “equals by definition”

2. Once the choice of bases is made, a linear mapping (also called a homomorphism) is represented by a matrix (see section 13.3.2). Abusing the language, we can identify this linear mapping with this matrix. And this is what we do here. “Abuse of language” (in the sense widely used in mathematics) and “abuse of notation” are considered to be synonymous in this book.

Chapter 2

Systems Theory (I)

The objective of this chapter is to specify the notion of system and to introduce a few basic concepts. Chapter 7 provides the complementary material that is necessary for a good understanding of the notions and methods presented in Chapters 3 to 6. In this present work, we are mainly interested in “linear time-invariant” systems (see section 2.2.5). The other types of systems will only be briefly mentioned.

2.1. Introductory example

Consider the RLC circuit in Figure 1.1 (). It is governed by . To simplify the calculations, let us assume 0. can be written as

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